I had someone tell me, after seeing what grade they got on several quizzes, that he got the highest grade (after rounding to the nearest percent) for that letter, for all 5 quizzes. Each quiz he took, he got a different letter grade on it.

When the grades are calculated for these quizzes, it is the number of questions you got right divided by the number of questions total on the quiz. Then the score is taken as a percent (the decimal is multiplied by 100) and then rounded to the nearest percent (.5 and above rounds up, below .5 rounds down)

The grading scale works so that:

100-90 A
89-80 B
79-70 C
69-60 D
0-59 F

What are the fewest number of questions possible on each quiz?

For example, if someone got 6 questions right out of 7 questions total, it would be 6/7 or about 85.7%, which rounds to 86, which isn't the highest B possible, (86 is not equal to 89) Since no number over 7 can be in the 88.5 up to 89.5 range, there couldn't have been exactly 7 questions on the quiz.
There also couldn't have been 8 questions on the B quiz. 7/8 or 88 percent isn't the highest B possible; it's too low, and 8/8 or 100 percent is too high.

If the problem read that each quiz had to have the same number of questions, the results are as follows:
Since it is possible to score exactly 100,89,79,69 and 59 on a 100-question quiz, the number of questions must be less than or equal to 100. Checking all possibilities by brute force results in:
61/61 = 100%
54/61 = 88.52% ~ 89%
48/61 = 78.69% ~ 79%
42/61 = 68.85% ~ 69%
36/61 = 59.02% ~ 59%